A primer for financial engineering

One needs to understand thepurpose, financial structure, and properties of such a financial instrument high-in order to study and model its behavior high-in time, high-intelligently pric

A Primer for Financial Engineering

A Primer for Financial

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ISBN: 978-0-12-801561-2

To Daria, Fred and Irma

To Tuba and my parents

This book presents the authors’ professional reflections on finance, ing their exposure to and interpretations of important problems historicallyaddressed by experts in quantitative finance, electronic trading, and riskengineering The book is a compilation of basic concepts and frameworks

includ-in finclud-inance, written by enginclud-ineers, for a target audience includ-interested includ-in pursuinclud-ing

a career in financial engineering and electronic trading The main goal ofthe book is to share the authors’ experiences as they have made a similartransition in their professional careers

It is a well recognized phenomenon on the Street that many engineers andprogrammers working in the industry are lacking the very basic theoreticalknowledge and the nomenclature of the financial sector This book attempts

to fill that void The material covered in the book may help some of them tobetter appreciate the mathematical fundamentals of financial tools, systems,and services they implement and are utilized by their fellow investmentbankers, portfolio managers, risk officers, and electronic traders of allvarieties including high frequency traders

This book along with [1] may serve as textbook for a graduate levelintroductory course in Financial Engineering The examples given in thebook, and their MATLAB codes, provide readers with problems and projecttopics for further study

The authors have benefited over the years from their affiliation with Prof.Marco Avellaneda of Courant Institute of Mathematical Sciences at the NewYork University Thank you, Marco

Ali N AkansuMustafa U TorunFebruary 2015

viii

as well as understanding the market microstructure (studies on modelingthe limit order book activity), then builds trading and risk managementstrategies using those models, and develops execution strategies to get inand out of investment positions in an asset The list of typical questionsfinancial engineers strive to answer include

• “What is the arrival rate of market orders and its variation in the limitorder book of a security?”

• “How can one partition a very large order into smaller orders such that itwon’t be subject to significant market impact?”

• “How does the cross correlation of two financial instruments vary intime?”

• “Do high frequency traders have positive or negative impact on themarkets and why?”

• “Can Flash Crash of May 6, 2010 happen again in the future? What wasthe reason behind it? How can we prevent similar incidents in the future?”and many others We emphasize that these and similar questions andproblems have been historically addressed in overlapping fields such asfinance, economics, econometrics, and mathematical finance (also known

as quantitative finance) They all pursue a similar path of applied study.Mostly, the theoretical frameworks and tools of applied mathematics,

A Primer for Financial Engineering http://dx.doi.org/10.1016/B978-0-12-801561-2.00001-0

1

statistics, signal processing, computer engineering, high-performance puting, information analytics, and computer communication networks areutilized to better understand and to address such important problems thatfrequently arise in finance We note that financial engineers are sometimescalled “quants” (experts in mathematical finance) since they practice quan-titative finance with the heavy use of the state-of-the-art computing devicesand systems for high-speed data processing and intelligent decision making

com-in real-time

Although the domain specifics of application is unique as expected, theinterest and focus of a financial engineer is indeed quite similar to what asignal processing engineer does in professional life Regardless of the appli-cation focus, the goal is to extract meaningful information out of observedand harvested signals (functions or vectors that convey information) withbuilt-in noise otherwise seem random, to develop stochastic models thatmathematically describe those signals, to utilize those models to estimateand predict certain information to make intelligent and actionable decisions

to exploit price inefficiencies in the markets Although there has been anincreasing activity in the signal processing and engineering community for

finance applications over the last few years (for example, see special issues

of IEEE Signal Processing Magazine [2] and IEEE Journal of SelectedTopics in Signal Processing [3], IEEE ICASSP and EURASIP EUSIPCOconference special sessions and tutorials on Financial Signal Processing

and Electronic Trading, and the edited book Financial Signal Processing and Machine Learning [1]), inter-disciplinary academic research activity,

industry-university collaborations, and the cross-fertilization are currently attheir infancy This is a typical phase in the inter-disciplinary knowledge gen-eration process since the disciplines of interest go through their own learningprocesses themselves to understand and assess the common problem areafrom their perspectives and propose possible improvements For example,speech, image, video, EEG, EKG, and price of a stock are all described assignals, but the information represented and conveyed by each signal is verydifferent than the others by its very nature In the foreword of Andrew Pole’sbook on statistical arbitrage [4], Gregory van Kipnis states “A descriptionwith any meaningful detail at all quickly points to a series of experimentsfrom which an alert listener can try to reverse-engineer the [trading] strategy.That is why quant practitioners talk in generalities that are only understand-able by the mathematically trained.” Since one of the main goals of financialengineers is to profit from their findings of market inefficiencies comple-mented with expertise in trading, “talking in generalities” is understandable

However, we believe, as it is the case for every discipline, financial neering has its own “dictionary” of terms coupled with a crowded toolbox,and anyone well equipped with necessary analytical and computational skillset can learn and practice them We concur that a solid mathematical training

engi-and knowledge base is a must requirement to pursue financial engineering

in the professional level However, once a competent signal processing

engineer armed with the theory of signals and transforms and computational skill set understands the terminology and the finance problems of interest,

it then becomes quite natural to contribute to the field as expected Themain challenge has been to understand, translate, and describe financeproblems from an engineering perspective The book mainly attempts tofill that void by presenting, explaining, and discussing the fundamentals,the concepts and terms, and the problems of high interest in financialengineering rather than their mathematical treatment in detail It should beconsidered as an entry point and guide, written by engineers, for engineers

to explore and possibly move to the financial sector as the specialty area.The book provides mathematical principles with cited references and avoidsrigor for the purpose We provide simple examples and their MATLABcodes to fix the ideas for elaboration and further studies We assume thatthe reader does not have any finance background and is familiar withsignals and transforms, linear algebra, probability theory, and stochasticprocesses

We start with a discussion on market structures in Chapter 2 We light the entities of the financial markets including exchanges, electroniccommunication networks (ECNs), brokers, traders, government agencies,and many others We further elaborate their roles and interactions in theglobal financial ecosystem Then, we delve into six most commonly tradedfinancial instruments Namely, they are stocks, options, futures contracts,exchange traded funds (ETFs), currency pairs (FX), and fixed incomesecurities Each one of these instruments has its unique financial structureand properties, and serves a different purpose One needs to understand thepurpose, financial structure, and properties of such a financial instrument

high-in order to study and model its behavior high-in time, high-intelligently price it, and

develop trading and risk management strategies to profit from its usually short lived inefficiencies in the market In Chapter 2, we also provide

the definitions of a wide range of financial terms including buy-side andsell-side firms, fundamental, technical, and quantitative finance and trading,traders, investors, and brokers, European and American options, initialpublic offering (IPO), and others

We cover the fundamentals of quantitative finance in Chapter 3 Eachtopic discussed in this chapter could easily be extended in an entire chapter

of its own However, our goal in Chapter 3 is to introduce the very basicconcepts and structures as well as to lay the framework for the followingchapters We start with the price models and present continuous- anddiscrete-time geometric Brownian motion Price models with local andstochastic volatilities, the definition of return and its statistical propertiessuch as expected return and volatility are discussed in this chapter Afterdiscussing the effect of sampling on volatility and price models withjumps, we delve into the modern portfolio theory (MPT) where we discussthe portfolio optimization, finding the best investment allocation vectorfor measured correlation (covariance/co-movement) structure of portfolioassets and targeted return along with its risk Next, Section 3.4 revisits thecapital asset pricing model (CAPM) that explains the expected return of

a financial asset in terms of a risk-free asset and the expected return ofthe market portfolio We cover various relevant concepts in Section 3.4including the capital market line, market portfolio, and the security marketline Then, we revisit the relative value and factor models where the return of

an asset is explained (regressed) by the returns of other assets or by a set offactors such as earnings, inflation, interest rate, and others We end Chapter 3

by revisiting a specific type of factor that is referred to as eigenportfolio asdetailed in Section 3.5.4 Our discussion on eigenportfolios lays the ground

to present a popular trading strategy called statistical arbitrage (Section 4.6)

in addition to filter the built-in market noise in the empirical correlationmatrix of asset returns (Section 5.1.4)

As highlighted in Chapter 4, the practice of finance, traders, and tradingstrategies may be grouped in the three major categories These groups arecalled fundamental, technical, and quantitative due to their characteristics.The first group deals with the financials of companies such as earnings, cashflow, and similar metrics The second one is interested in the momentum,support, and trends in “price charts” of the markets Financial engineersmostly practice quantitative finance, the third group, since they approachfinancial problems through mathematical and stochastic models, implement-ing and executing them by utilizing the required computational devices andtrading infrastructure

In contrast to investing into a financial asset (buying and holding a rity for relatively long periods), trading seeks short-term price inefficiencies

secu-or trends in the markets The goal in trading is simple It is to buy low and

sell high, and make profit coupled with a favorable risk level Professionaltraders predefine and strictly follow a set of systematic rules (tradingstrategies) in analyzing the market data to detect investment opportunities

as well as to intelligently decide how to react to those opportunities InChapter 4, we focus on quantitative (rules based) trading strategies First, wepresent the terminology used in trading including long and short positions,buy, sell, short-sell, and buy-to-cover order types, and several others Weintroduce the concepts like cost of trading, back-testing (a method to test

a trading strategy using historical data), and performance measures for atrading strategy such as profit and loss (P&L) equitation and Sharpe ratio.Then, we cover the three most commonly used trading strategies The firstone is called pairs trading where the raw market data is analyzed to look forindicators identifying short lived relative price inefficiencies between a pair

of assets (Section 4.5) The second one is called statistical arbitrage wherethe trader seeks arbitrage opportunities due to price inefficiencies acrossindustries (Section 4.6) The last one is called trend following where onetracks strong upward or downward trends in order to profit from such a pricemove (Section 4.7) In the latter, we also discuss common trend detectionalgorithms and their ties to linear-time invariant filters At the end of eachsection, we provide recipes that summarize the important steps of the giventrading strategy In addition, we also provide the MATLAB implementations

of these strategies for the readers of further interest We conclude thechapter with a discussion on trading in multiple frequencies where tradersgain a fine grained control over the cycle of portfolio rebalancing process(Section 4.8)

Return and risk are the two inseparable and most important performance

metrics of a financial investment It is quite analogous with the two

inseparable metrics of rate and distortion in rate-distortion theory [5] In

Section 3.3.1.2, we define the risk of a portfolio in terms of the correlation

matrix of the return processes for the assets in the portfolio, P For a

portfolio of N assets, there are N (N − 1)/2 unknown cross-correlations

and they need to be estimated through market measurements in order

to form the empirical correlation matrix, ˆP It is a well-known fact that

ˆP contains significant amount of inherent market noise In Chapter 5,

we revisit random matrix theory and leverage the asymptotically knownbehavior of the eigenvalues of random matrices in order to identify thenoise component in ˆP, and utilize eigenfiltering for its removal from

measurements Later in the chapter, we extend this method for the portfoliosformed with statistical arbitrage (or any form of strategy that involves

hedging of assets) (Section 5.1.4) and also for the case of trading in multiplefrequencies (Section 5.2) The chapter includes various advanced methodsfor risk estimation (Section 5.3) like Toeplitz approximation to the empiricalcorrelation matrix ˆP and use of discrete cosine transform (DCT) as an

efficient replacement to Karhunen-Loéve transform (KLT) in portfolio riskmanagement Once the risk estimation is performed, the next step is tomanage the risk The main question to be addressed is “given that we knowhow to estimate the risk for the given investment allocation vector andcorrelation matrix of asset returns, how do we make a decision to change ourpositions in the assets in order to keep the investment risk at a predefinedlevel?” We conclude the chapter with discussions on three different riskmanagement methods They are called stay in the ellipsoid (SIE), stay onthe ellipsoid (SOE), and stay around the ellipsoid (SAE) as described in

Section 5.4 (The locus of q i, 1≤ i ≤ N satisfying (3.3.7) for a fixed value

of riskσp is an ellipsoid centered at the origin Hence, the names of these

three methods include the word ellipsoid.)

Regardless of the trading strategy and risk management method utilized

to come up with intelligent decisions on the investment allocation vector

in time to rebalance the portfolio, the last and very important piece of thepuzzle is to place orders and have them executed as originally planned Themarket and limit order types are described in Chapter 6 The effect of orders

on the current market price of an asset, called the market impact, and severalalgorithmic trading methods to mitigate it are discussed in this chapter.Then, we delve into market microstructure, examining the limit order book(LOB) and its evolution through placement of limit and market orders Afterrevisiting and commenting on important phenomenon in finance called Eppseffect, the drop in the measured pairwise correlation between asset returns asthe sampling (trading) frequency increases, we make some remarks on highfrequency trading (HFT) We survey through publicly known HFT strategiesand highlight the state-of-the-art on covariance estimation techniques withthe high frequency data We conclude the chapter with discussions on thelow-latency (ultra high frequency) trading and impact of technology centricHFT on the financial markets

The concluding remarks are presented in Chapter 7, followed by areference list of books and articles cited in the book that may help the readers

of interest for further study

1.1 DISCLAIMER

The authors note that the material contained in this book is intended onlyfor general information purpose and is not intended to be advice on anyinvestment decision The authors advise readers to seek professional advisebefore investing in financial products The authors are not responsible orliable for any financial or other losses of any kind arising on the account ofany action taken pursuant to the information provided in this book

CHAPTER 2

Financial Markets and Instruments

2.1 STRUCTURE OF THE MARKETS 8 2.2 FINANCIAL INSTRUMENTS 12 2.3 SUMMARY 21

We start with the definitions and descriptions of different entities in thefinancial markets and how they interact with each other We introduceexchanges, electronic communication networks (ECNs), broker-dealers,market-makers, regulators, traders, and funds to better understand the finan-cial ecosystem given that the legal framework, regulatory and complianceissues are beyond the scope of this book In a separate section, we delve intothe details of various types of financial instruments, i.e., stocks, options,futures, exchange traded funds (ETFs), currency pairs, and fixed incomesecurities Our goal in this chapter is not to provide the exhaustive details

of these entities, instruments, and their relationships with each other Werather aim to equip engineers with a good understanding of these concepts

to navigate further in the area they choose to focus on through the referencesprovided We note that our primary focus is the financial markets andproducts offered in the United States However, with only slight nuances,the concepts and definitions are globally applicable to any local financialmarket of interest

2.1 STRUCTURE OF THE MARKETS

Financial instruments are bought and sold in venues called exchanges There

are many exchanges around the world Most of them are specific to aparticular class of financial instrument New York Stock Exchange (NYSE),London Stock Exchange, and Tokyo Stock Exchange are some of the major

stock exchanges around the world Chicago Mercantile Exchange (CME),

Chicago Board of Trade (CBOT), and London International Financial

Fu-tures and Options Exchange (LIFFE) are the largest fuFu-tures exchanges in the

world Similarly, Chicago Board Options Exchange (CBOE) is the largest

options exchange in the world Exchanges are essentially auction markets in

8 A Primer for Financial Engineering http://dx.doi.org/10.1016/B978-0-12-801561-2.00002-2

where parties bid and ask to buy and sell financial instruments, respectively Traditionally, traders (people trading in their own accounts) and brokers

(people trading in their clients’ accounts) trade stocks and other instruments

by being physically present in the exchange building, on the exchange floor

Hence, they are called floor traders and floor brokers There are people on the exchange floor called specialists who are responsible for facilitating

the trades, i.e., building the order book and matching the orders as well

as maintaining liquidity of the product (availability of the financial product

for buying and selling) When an investor wants to buy or sell a particular

financial product, they call their banks or their brokers and place the order Then, floor brokers are notified and they execute the order through the specialist, and send back the confirmation.

However, over time, the trading infrastructure and procedures havesignificantly changed with the introduction of ECNs They are specialcomputer networks for facilitating the execution of trades and carryingreal-time market information to their consumers ECNs provide access toreal-time market data and let brokers and large traders trade with each

other eliminating the need for a third party through direct market access

(DMA) Some of the largest ECNs are Instinet, SelectNet by NASDAQ(National Association of Securities Dealers Automated Quotations), andNYSE Archipelago Exchange (NYSE ARCA)

The exchanges we have cited so far are very large in volume althoughthey are just the tip of the iceberg in terms of the number of tradingvenues around the globe Naturally, there is inter-exchange trading activity,flow of quotes for bids and asks for various types of products amongthose national and global venues Just like exactly the same product mayhave different prices and sales volumes at two different grocery stores, thesame financial product is usually traded at a different price, with different

volume and liquidity, at different trading venues This fragmented market

structure makes it hard for small investors to trade at the best venue due to

their lack of scale Broker-dealers (BDs) facilitate trades for their clients in

multiple venues through their sophisticated infrastructure comprised of thestate-of-the-art low latency (high speed) data networking, high performancecomputing and storage facilities Instead of having an account in each venueand build a costly trading infrastructure, the investor has one account withhis broker that consolidates and offers most of these services for a fee.Brokers promise their clients to have their orders executed at the bestprice available in the market for a pre-defined transaction cost Brokerscompete among themselves by offering their clients low transaction costs,

high liquidity, and access to many venues with lowest possible latency.Some brokers also provide sampled or near real-time market data to their

clients for use in their algorithmic trading activity.

Investors, small or large scale, who randomly appear and disappear are

not the only source of liquidity in the market Market makers simultaneously

place large buy and sell orders to maintain price robustness and stability in

the limit order book (LOB) of a stock Whenever there is an execution in

either side, they immediately sell or buy from their own inventory or find

an offsetting order to match Market makers seek to profit from the price difference between the bid and ask orders they place that is called the spread.

Since the size of the orders they place is usually much larger than the order

size of a typical trader in the market, they in a sense, define, or make the market for that particular instrument.

Activity in the financial markets is regulated by government and governmental agencies In the United States, the leading agency is the

non-Securities Exchange Commission (SEC) which has the primary

respon-sibility to enforce securities laws as well as to regulate the financialmarkets in the country Moreover, there are agencies specific to a market

or a small number of markets, such as the Commodity Futures Trading Commission (CFTC) that regulates the futures and options markets In

other countries, there are similar governmental and non-governmental

bodies for the task International Organization for Securities Commission

(IOSCO) is responsible to regulate the securities and futures markets aroundthe globe

We have discussed, without any details, the basic concepts of facilitationand regulation of a trade Although the main incentive to trade a financialinstrument is to make profit by buying low and selling high, there aredifferent types of players in the market based on their motivations to exploit

the asset price inefficiencies and the way they trade Buy-and-hold investors

seek to profit in long term by holding the financial instrument for a longtime, potentially years, and sell when the price is significantly high, leading

to a high profit per trade over long time On the other hand, speculators are

not interested in holding the instrument for a long time and they seek toaggregate big profit from small profits on very large number trades In the

extreme case, there are high frequency traders in the market Their holding

times are as low as under a second or much less Readers more interested are

referred to Section 6.4 for a detailed discussion of high-frequency trading

(HFT) There are also different types of processes and methods used to

make a decision to trade a particular asset Fundamental traders, technical traders, and quantitative traders (quants) employ different techniques to

reach a decision to buy or sell an asset or a basket of assets We presentthe foundations of these three trading categories in Chapter 4

Only a small percentage of people manage their own money in themarket through single or multiple investment accounts with broker-dealers

They are called individual investors In contrast, most of the investors enter the market through investment funds Funds are managed by finance

professionals and serve as an investment vehicle for small and institutionalinvestors The fund managers make investment decisions on behalf of theirclients through extensive financial analysis risk-return trade-offs combinedwith market intelligence and insights Those investors who want to investbut do not know how to analyze the market and relevant risks of an

investment commonly become a client of a mutual fund or a hedge fund In

general, mutual funds manage large amounts of investment capital, investing

on behalf of many small and big investors On the other hand, in general,hedge funds have smaller amount of capital under management invested by alower number of investors They usually invest in a wide variety of financialinstruments Traditionally speaking, hedge funds almost always employhedged investment strategies to monitor and adjust the risk However,today, a hedge fund often utilizes a large number of strategies comprised

of hedged and non-hedged ones Hedge funds are not open to investmentfor the general public Therefore, they are less regulated compared to otherfunds

It is quite common in the financial sector to refer to different marketparticipants, e.g., investment banks and firms, analysts and institutions, as

buy side and sell side entities Basically, the difference between the two sides is about who is buying and who is selling the financial services.

Sell side entities provide services such as financial advice, facilitation

of trades, development of new products such as options (Section 2.2.2),futures contracts (Section 2.2.3), ETFs (Section 2.2.4), and many others.Such entities include investment banks, commercial banks, trade executioncompanies, and broker-dealers In contrast, buy side entities buy thoseservices in order to make profits from their investments on behalf of theirclients Some of these participants are hedge funds, mutual funds, assetmanagement companies, and retail investors

In the next section, we present the details of various types of mostcommonly traded financial instruments and highlight their specifics

2.2 FINANCIAL INSTRUMENTS

We cover six types of the most popular financial instruments in this section.Namely, they are publicly traded company stocks, options, futures contracts,exchange traded funds (ETFs), currency pairs, and fixed income securities

We note that this list of instruments and our coverage of them are notexhaustive However, this section will equip us with sufficient detail tounderstand different types of financial products and their distinctions

2.2.1 Stocks

A stock represents a fraction of ownership in a company It is delivered in the units of shares However, it is also common to use the word “stock” rather than “stock share.” Owner of a stock is called a shareholder of the company.

A shareholder owns a percentage of the company determined by the ratio ofthe number of shares owned to the total outstanding shares It is common

that a company offers two types of stocks Namely, they are common stock and preferred stock A common stock entitles the owner a right to vote

in the company whereas a preferred stock does not However, a preferred

stock has higher priority in receiving dividends and when the company

files for bankruptcy A dividend is a fraction of the profits delivered to theshareholders Companies, as they make profit, may pay dividends to theirshareholders throughout a year Dividends can be paid in cash or in stocks.Every stock is listed in a specific exchange such as NYSE or NASDAQ.However, they can be traded in multiple exchanges, ECNs, and possibly

in dark pools as well In order to identify stocks and standardize the naming

method, every stock has a symbol or ticker For example, the ticker for Apple Inc stock is AAPL and the ticker for Google Inc stock is GOOG Moreover,

there might be additional letters in the ticker of a stock, commonly separated

by a dot, in order to differentiate different types of stocks (common,preferred, listed in US, listed in Mexico, etc.) a company offers to investors.For example, Berkshire Hathaway Inc offers two different common stockswith the tickers BRK.A and BRK.B with their specific privileges

Companies issue their stocks for the first time through an initial public offering (IPO) process An IPO is assisted by an underwriter, a financial institution facilitating the IPO process IPO is also referred to as going public since the company is not privately held anymore and its shares are

publicly traded A publicly traded company has to release a report on itsfinancials and major business moves every quarter, and is subject to strict

regulations Therefore, there are still some very large companies that areprivately held Whether small or large, companies go public usually to raisecash, have liquidity in the market, offer stock options to their employees andattract talent, improve investor trust in them that may lead to reduced interestrates when they issue debts (Section 2.2.6), etc Once publicly available,stocks can be purchased directly from the company itself, directly from

an exchange via direct market access, or through a broker-dealer (mostcommon)

In the old days, being listed in a major stock exchange was veryimportant and prestigious for a company However, it is changing as highernumber of startups are going public Stocks of those highly reputable and

reliable companies with large market capitalization (market cap), the total value of the outstanding shares in the market, are sometimes called blue chip

stocks (a reference to poker game) On the other extreme, stocks that do notmeet the minimum criteria to be listed in stock exchanges are traded in the

over-the-counter (OTC) markets These stocks are called as pink-sheets (a reference to the color of the certificate) or penny stocks.

An index is the sum, or weighted sum, of stock prices for a basket (group)

of stocks They are mathematical formulas defining certain benchmarks

to track market performance and they cannot be traded The list of mostwidely quoted indices include Standard & Poor 500 (S&P 500), Dow JonesIndustrial Average (DJIA), NASDAQ Composite (NASDAQ 100), Russell

1000, FTSE Eurotop 100, DAX, and Nikkei 225 There are many otherindices with their focuses

When the price of a stock gets very high, it is harder for small investors

to buy and sell the stock, leading to reduced liquidity and possibly largerbid-ask spread (Section 6.2) In order to attract more liquidity, a company

issues a stock split For example, a 3-for-1 split delivers three new shares for each old share and simultaneously reducing the stock price by a factor

of 3 Similarly, if the price of a stock goes very low, a company may issue a

reverse split As an example, a 1-for-4 reverse split delivers 1 new share for four old shares that increases stock price by a factor of 4 We note

that splits do not change the market capitalization of a stock Splits anddividends reveal themselves as impulses (jumps) in stock price and stockvolume time series Therefore, instead of using the raw historical price andvolume data of a stock, it is preferred to use the adjusted price and volumethat accounts for the splits and dividends InFigure 2.2.1 we see closingprice and adjusted price for Apple Inc (APPL) stock

Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug

2.2.2 Options

An option is the right (not an obligation) to buy or sell a financial instrument

at predefined price and terms Underlying instruments may be stocks, bonds,market indices, commodities, bonds, and others In this section, we focus on

stock options for brevity and simplicity In general, option contracts may be

quite sophisticated and complex based on their design objectives The buyer

of a call option (call) has the right to buy the underlying stocks at the strike price and the seller of the call is obligated to deliver stocks if the buyer decides to exercise the option on or before the expiration date Similarly, the buyer of a put option (put) has the right to sell the underlying stock at the strike price and the seller of the put is obligated to buy the stock if the

buyer decides to exercise this right on or prior to its expiration Although thetransaction can be facilitated by a third party, option is a contract betweenthe two parties, the seller and the buyer In a sense, the seller writes a contract(legal document) and creates a tradable financial instrument Therefore,

seller of an option is also called the writer of the option Options have their expiration (maturity) dates A European option can be exercised only on its expiration date whereas an American option may be exercised at any time.

In general, an option structured in US markets represents right to buy or sell

100 shares of the underlying stock

Naturally, an option is not for free and it has its initial cost Let S, K, and C be the spot price, strike price, and initial cost of buying an option,

respectively A long position in a call option (long call) pays off, call is

in-the-money, only when S > K + C Moreover, long call is at-the-money

when S = K + C, and out-of-the-money when S < K + C Similarly, a long put is in-the-money when S < K − C, at-the-money when S = K − C, and out-of-the-money when S > K − C The payoff curves for long call,

long put, short call (a short position in a call option), and short put aredisplayed inFigure 2.2.2 The buyer and the seller of the option is said to

be in a long and short position in the option, respectively We observe from

Figure 2.2.2 that long call, long put, and short put have limited downsidewhereas downside for short call is unlimited Similarly, all positions but the

long call has limited upside Long call and short put are bullish positions whereas long put and short put are bearish positions.

Options are mainly used for hedging and leverage of an open position

in order to adjust the total risk An investor can hedge his long position in

a stock, to reduce the risk of his long position, by buying a put option onthe same stock Therefore, at the expiration date of the option, if the market(spot) price of the stock is less than the strike price of the option, investorcan exercise the option and sell the stocks at a higher price than the spotprice, hence, reduce his loss Moreover, an option of a stock is usually muchcheaper than its spot price Therefore, instead of buying the shares of a stock,

an investor can buy a larger number of put options with the same capital Inthis scenario, spot price of the stock still needs to go up for the investor to

profit However, the investor can generate much higher profit by exercisingthe large number of put options, rather than selling the stocks itself, hence,leveraging their capital.

Speculators in the market use combinations of the basic four positions incalls and puts and basic two positions in stocks to tailor investment strategiesfor a given risk target The list of well-known strategies include the coveredput, covered call, bull spread, bear spread, butterfly spread, iron condor,straddle, and strangle For example, in a butterfly spread, an investor buys

a call at strike price K1, sells two calls at K2, and sells a call at K3 where

K3> K2> K1all with the same expiration date Butterfly spread allows the

speculator to profit when the spot price is close to K2at the expiration date,and limits their loss otherwise

Given their increased flexibility and complexity, options pricing is muchmore involved than stock pricing, and it has a very rich literature to learnfrom In general, valuation models for options depend on the current spotprice of the underlying asset, strike price, time to expiration date, and thevolatility of the asset Fischer Black and Myron Scholes developed theircelebrated closed-form differential model for the price of a European option

known as the Black-Scholes options pricing formula [6] According to the formula, prices of a call and put option with spot price S, strike price K, and time-to-expiration date t are defined as



S K

+



r+σ22



t



d2 d1− σ√t,

and σ is the volatility of the underlying asset We note that d1 and d2

are nothing else but defined parameters for ease of notation The Scholes model is often critiqued for impracticality since it assumes constantvolatility and interest rate in time Moreover, it is not easy in the model toaccount for the dividends paid for the underlying asset However, it forms

Black-the basis for many oBlack-ther Black-theoretical models including Black-the Heston model [7]

in which the volatility is not constant in time but a stochastic process Geske-Whaley method is used to solve the Black-Scholes equation for an

Roll-American option with one dividend [8] Binomial options pricing modeldeveloped by John Cox, Stephen Ross, and Mark Rubinstein known as

the Cox-Ross-Rubinstein (CRR) model is used to model the price in the

form of a binomial tree with each leaf corresponding to a possibility of thevalue at the expiration date Granularity in CRR model can be adjusted

to closely approximate the continuous-time Black-Scholes model CRR

is usually preferred over Black-Scholes model since it can model bothEuropean and American options and dividend payments with ease in thetree Nevertheless, for most of the option classes, the differential equations

in the models are intractable In this case, one resorts to Monte Carlosimulation and finite difference methods to price the options

2.2.3 Futures Contracts

A forward contract is an agreement between a seller and a buyer to deliver and purchase, respectively, a particular amount of an asset, an underlying,

at a predefined price (delivery price) at a predefined date, delivery date.

Traditionally, the underlying is a commodity such as wheat, oil, steel, silver,coffee beans, cattle, and many other tradable assets However, underlyingassets also include financial instruments such as currencies, treasury bonds,interest rates, stocks, market indices, and others There are futures evenfor weather conditions, freight routes, hurricanes, carbon emissions, andmovies

Historically, forward contracts were used for balancing the supply anddemand of the agricultural products such as wheat and cotton Farmerswould optimize their crops based on the contracts they have and thebuyers would plan accordingly as both parties had a mutual contract

to deliver/receive a product at a certain price at a certain date, hence,reducing their risks Over time, this market had grown dramatically turning

into a global futures market The difference between a forward contract and a futures contract is that the latter is standardized, regulated, mostly

traded in the exchanges, and cleared by financial institutions Two largeexchanges for futures are the Chicago Board of Trade (CBOT) and theLondon International Financial Futures and Options Exchange (LIFFE) Inthe Unites States, the futures market is regulated by the Commodity FuturesTrading Commission (CFTC) and the National Futures Association (NFA).The futures market is very liquid that makes it an attractive place tohedge investments and speculate their prices There are still participants inthe futures market that seek to buy and sell contracts to reduce their risk,

hedge, for a particular commodity However, similar to the options case, in

today’s global futures market, most of the trading activity is generated from

the speculation A great majority, almost all, of the futures contracts end

without physical delivery of their underlying assets Common speculationstrategies are similar to those for options; going long, going short, or buying

a spread of contracts Typical spreads are calendar, inter-underlying, andinter-market in which the speculator buys and sells contracts for the sameunderlying with different delivery dates, for different underlying assets withthe same expiration date, and for the same underlying and same expirationdate in different exchanges, respectively

In the futures markets, positions are settled every day Therefore, in order

to trade in the futures market, one needs a margin, cash in a bank account,for daily credits and debits due to potential gains and losses due to marketfluctuations However, margin requirements are relatively low, hence the

leverage is high Therefore, for the same amount of capital, one can bet for

a larger number of underlying assets by buying the futures contract of theasset, rather than buying the asset itself

Pricing of the futures contracts is fundamentally done by the assumption

of no arbitrage Let the delivery price, forward price, of a contract to be

F (T) and current spot price to be S(t) where T is the time to delivery date Then, the payoff for the seller of a contract is F (T) − S(T) at the delivery However, seller can cash-and-carry the underlying asset Namely, he can own one unit of the asset and have a debt of S (t) dollars, with the payoff amount S (T) − S(t)e r (T−t) at the delivery where r is the interest rate Hence,

the total payoff of the seller is expressed as

F(T) − S(T) + S(T) − S(t)e r(T−t) = F(T) − S(t)e r(T−t)

Due to the no arbitrage assumption, we have zero payoff Therefore, theforward price of a contract is equal to

F(T) = S(t)e r (T−t).

2.2.4 Exchange Traded Funds (ETFs)

ETF is a financial product that tracks a basket of other financial instruments

or an index They are similar to a mutual fund, however, they are tradedjust like a stock in the exchanges Therefore, it is possible to invest in abasket or an index fund with very little capital or short-sell the fund tobet against an index ETFs allow investors to diversify their portfolios withlittle cost Namely, the only cost involved is the transaction cost paid to the

broker-dealers For example, in order to invest in the tech sector, intuitively,

we would need to buy at least three large tech stocks such as Apple Inc.(AAPL), Microsoft Corporation (MSFT), and Google Inc (GOOG) Ideally,

we would need to buy all the stocks listed in the NASDAQ-100 index,

a weighted average of the top 100 technology stocks based on marketcapitalization Instead, we can simply buy PowerShares QQQ ETF whichtracks the NASDAQ-100 index Some ETFs are actively managed, theirmanagers actively seek to beat the market Holder of a managed ETF is

charged a management fee In general, an ETF has much smaller expense

ratio than a mutual fund The specifics of the basket and management feefor an ETF are found in its prospectus available by their issuers

Net asset value (NAV) of a mutual fund is calculated at the end of each

day whereas the price of an ETF changes in real time during the day as it

is traded just like a stock in the exchanges Therefore, they are also heavilyused for speculation However, buy and hold investors still gain performanceclose to the underlying index or basket as ETFs are rebalanced periodically

to match performance over a long term For example, at the time of writing,according to its prospectus, portfolio of QQQ is rebalanced quarterly andreconstituted annually

Leveraged ETFs are higher risk instruments designed to match the twice

or triple performance of an index There are also short leveraged ETFs betting against indices They target to match the twice or triple opposite performance of an index The former and latter are useful for investors who

believe the market is going to go up and down, respectively, and leveragetheir bets aggressively However, it is common that some leveraged ETFsfail to deliver their predicted returns See [9] for a detailed study on thepath-dependence of the leveraged ETFs

2.2.5 Currency Pairs

Foreign exchange markets, often abbreviated as forex or FX, are the largest

in trading volume and capital in the world Participants of these marketstrade pairs of currencies by simultaneously buying a currency and sellingthe other A simple example is the US Dollars and Japanese Yen pair(USD/JPY) In this example, USD is the base currency and the JPY is thecounter currency A quotation for USD/JPY as 109.34 means that 1 USD isequal to 109.34 JPY In addition to the rate, the spread of a currency defined

as the rate difference between buying and selling it is important Spread

is measured in pips The pips is the smallest change for a rate It usually

is one percent of a percent For example, a pip of 1 USD is 0.0001 USD.Brokers in FX market profit from the spread since there are no commissions.Moreover, the currency trader pays and receives interest on the currencysold and bought, respectively The interest rate depends on the local countrythe traded currency belongs to Therefore, the difference in interest ratesfor both legs of the currency pair is also an important factor in making FXtrading decisions.

FX markets are decentralized and much less regulated compared toother markets due to their inherent cross-border market structure However,currency brokers in the United States must be registered with the FuturesCommission Merchants (FCMs) and they are regulated by CTFC Cur-rencies are traded worldwide The majority of the trades occur in Londonand New York The trading window is 24 hours during the business days.There is always high activity in some parts of the world during a day as thetrading starts with Asia, moves on through Europe, North America, and back

to Asia

Participants in the FX markets use fundamental analysis, evaluatingmacro-economical and political factors as well as gross domestic product,manufacturing and sales, consumer price index of the countries and othereconomic factors to make trading decisions Just like in stock markets

or derivative markets, speculators also use technical analysis tools such

as pivot points and Elliot waves for their trade decisions Nevertheless,financial engineers and quants participating and performing research on FXmarkets apply their prior knowledge and experiences from the stock andderivative markets and develop more sophisticated arbitrage models tailored

or tweaked for the currency pairs or a basket of currency pairs [10]

2.2.6 Fixed Income Securities

A fixed income security aims to deliver deterministic and mostly periodic

returns Bonds and money market securities are the most common types offixed income securities Investment in them brings very little risk (or no risk

at all) but also delivers relatively low return compared to other financialinstruments such as stocks An entity like a government, a municipalauthority, or a corporation issues bonds to lenders in order to borrow theircapital with pre-defined terms and conditions The issuer pays fixed interest

to the lender, usually every 6 months, until the maturity date of the bond.Money market securities are considered as the cash market or short-termsecurity market since they usually mature much faster than bonds

Money market securities include but are not limited to treasury bills(T-Bills), certificates of deposit (CD), commercial papers, and repurchaseagreements (repos) T-Bills are a way for governments to raise moneyfrom the public Price for a T-Bill is less than its face (par) price When

it matures, government pays the full face price to the holder Therefore,the deterministic return is the difference between the face value and thepurchase value Like bonds, T-Bills are generally purchased through afinancial bidding process They are considered as risk-free since they arebacked by a government However, the interest rate is often quite low CDsare similar to T-Bills but they are offered by private banks The interestrate is usually competitive but it is higher than that of a T-bill due to theincreased risk of default However, in the United States, CDs are insured,with certain limits, by Federal Deposit Insurance Corporation (FDIC) if thebank is a participant Commercial papers are similar but they are offered by

a corporation not a bank By issuing commercial papers, companies raisecash quickly and avoid the banks

In comparison to other markets, such as stock markets, the risk of moneymarkets and bonds, hence, their return, is very little Moreover, usually oneneeds larger capital to participate in the money market Therefore, most ofthe players in the money market are actually subscribers of large mutualfunds In the stock market, brokers do not hold any risk due to the openpositions created through them The capital move is between the investorand the exchange Brokerage business is merely to collect commissions asfacilitators to access financial markets through exchanges However, in themoney market, there are no exchanges Therefore dealers take the risk ontheir own account

2.3 SUMMARY

There are many venues around the world where financial instruments are fered to investors and traded by market participants Exchanges are auctionmarkets in which different parties bid and ask to buy and sell instruments,and orders are matched by specialists ECNs are special computer networkswith state-of-the-art data processing power that allow broker-dealers andlarge traders to create a market ecosystem to interact and trade directlyamong themselves In an ECN, the book keeping and order matching isdone by machines rather than humans, mostly in sub-milliseconds if not

of-in microseconds Market makers with certaof-in responsibilities and privileges

in the exchange place large orders simultaneously bidding and asking forthe same instrument They seek profiting from the spread between the bidand ask prices Other players in the market benefit from market makerssince they provide liquidity Financial markets are regulated mostly bygovernment agencies There are different types of investors based on theirasset holding times Namely, they are buy-and-hold investors, speculators,and high-frequency traders Another category for traders is based on themethods they utilize in order to make their trade decisions These tradertypes are called fundamental, technical, and quantitative traders Sell sideentities provide services to the buy side entities that make investmentdecisions on behalf of their clients in order to make profit by over-performing the market.

A stock represents a fraction of ownership in a company An option is theright (not an obligation) to buy or sell a financial instrument at predefinedprice and time A futures contract is an agreement between a seller and abuyer to deliver and to purchase, respectively, a particular amount of anasset at a predefined price and date An exchange traded fund (ETF) is

a financial product that tracks a basket of other financial instruments or amarket index providing investors a low-cost diversification Currency pairsare the largest contributors of trading activity where participants are seekingprofit by exploiting the spread between two foreign currencies A fixedincome security delivers deterministic and mostly periodic returns

CHAPTER 3

Fundamentals of Quantitative Finance

3.1 STOCK PRICE MODELS 23 3.2 ASSET RETURNS 28 3.3 MODERN PORTFOLIO THEORY 34 3.4 CAPITAL ASSET PRICING MODEL 38 3.5 RELATIVE VALUE AND FACTOR MODELS 46 3.6 SUMMARY 50

In this chapter, we revisit fundamental topics in quantitative finance ing continuous- and discrete-time stock price models; stock price modelswith jumps; return, expected return, volatility, Sharpe ratio, and cross-correlation of assets; portfolio optimization, modern portfolio theory, andcapital asset pricing model; relative value models, factor models, and eigen-portfolios We discuss these concepts from a signal processing perspectivealong with several others This chapter not only helps us to understandthe fundamentals but also prepares us for the discussions presented in thefollowing chapters

includ-3.1 STOCK PRICE MODELS

We briefly discussed the fundamental price models for options and futurescontracts in Sections 2.2.2 and 2.2.3, respectively In this section, we discuss

in detail the fundamental models for the price of stocks

3.1.1 Geometric Brownian Motion Model

Brownian motion, first discussed by Brown in 1827 in the context of motion

of pollens, further explained by Einstein in 1905, and formulated by Wiener

in 1918 has strong ties with the modeling of stock prices Bachelier in 1900described the price variation of a stock as a Brownian motion written as [11]

where t ≥ 0 is the independent time variable, p(t) is the price of the stock with an initial value p (0), σ is the volatility, μ is the drift, and w(t) is a Wiener process or standard Brownian motion such that dw (t) is a zero-mean

A Primer for Financial Engineering http://dx.doi.org/10.1016/B978-0-12-801561-2.00003-4

23

and unit-variance Gaussian process, i.e., dw (t) ∼ N (0, 1) However, this

model has problems First, according to the model, it is possible for price

to go below zero although stock prices are always positive Moreover,according to the model, change in price over a time period is not a function

of the initial price, p (0) It suggests that stocks with different initial prices

can have similar gains or losses in the same time interval This is not thecase in reality For example, probability of observing a $1 change in priceover a day is less for a stock priced at $10 than it is for a stock that is worth

$1,000 A better model for the stock price is the geometric Brownian motion

in which the rate of return for a stock is defined as

Before we discuss the proof, we provide three more definitions The

expected value and variance of p (t) are expressed as, respectively,

var [p (t)] = p2(0)e2μt

eσ2t− 1 (3.1.5)The log-price is defined as

We do not display the independent variables X (t) and t of functions

α [X(t), t], β [X(t), t], and f [X(t), t] in (3.1.9) for ease of notation Let

Since dw (t) ∼ N (0, 1), an infinitesimal increment in the log price (3.1.11)

is a Gaussian with mean

μ − σ2/2 dt and variance σ2dt Since summation

of the Gaussian random variables are also Gaussian it follows from (3.1.11)

that ln p (t) − ln p(0) is distributed as Gaussian with meanμ − σ2/2 t and

varianceσ2t Therefore, we write

μ − σ2/2 t + σ w(t) and jω  1 We know that μ − σ2/2 t + σ w(t)

is distributed as Gaussian with mean 

μ − σ2/2 t and variance σ2t, i.e.,

From (3.1.14) and (3.1.16), we have

3.1.2 Models with Local and Stochastic Volatilities

Geometric Brownian motion price model assumes a constant deviation of

a stock return, i.e., constant volatility, σ That assumption is not always

realistic since the markets and prices of stocks are affected by variousevents that occur randomly and may last for a long time in some cases,e.g., an economical crisis, or appear and vanish within minutes, e.g., the

Flash Crash of 2010 [14] (Section 6.4.4) Improved price models with local

[15, 16] and stochastic [7, 17] volatilities take into account that the volatilityitself is a function of time In models with local volatility, the price modelgiven in (3.1.2) is modified as

dp (t)

p (t) = μdt + σ (t) dw1(t), (3.1.22)where σ (t), i.e., volatility as a function of time, is a random process and

dw1(t) is a normal process One of the popular stochastic volatility models

is the Heston [7] model in which volatility is a random process that satisfiesthe stochastic differential as given

d σ (t) = κ [θ − σ (t)] dt + γσ (t)dw2(t), (3.1.23)whereκ is the mean-reversion speed, θ is the volatility in the long-term, γ is

the volatility of the volatilityσ (t), and dw2(t) is a normal process correlated with dw1(t) given in (3.1.22) According to the Heston model, volatility

is a mean-reverting process with a constant volatility and its infinitesimalchanges are related to the ones of the price We note that the model given in(3.1.23) is related to the celebrated Cox-Ingersoll-Ross process [18] used tomodel the short-term interest rates

3.1.3 Discrete-Time Price Models and Return

It is a common practice to sample the price in time and to refer the stockreturns with respect to their sampling periods, e.g., 30-min returns, 1-hreturns, and end of day (EOD) returns Discrete-time analog of geometricBrownian motion model is obtained by sampling the price with a certaintime period as given

s(n) = s(n − 1) + μ + σ ξ(n), (3.1.24)

where s (n) = ln p(n) is the log-price of a stock at discrete-time n with price

p (n), μ and σ are the drift and volatility of the stock, respectively, and ξ(n)

is the white Gaussian noise withξ(n) ∼ N (0, 1) The log-return at time n is defined as

in (3.1.25), g (n), due to the Taylor series expansion of the logarithm, i.e.,

Since the value of return might get very small, it is customary in finance to

use basis points (bps) instead of percent One bps is one percent of a percent,

i.e., 1 bps= 0.01%

3.2.1 Expected Return, Volatility, and Cross-Correlation

of Returns

Mean and standard deviation of the return of an asset,μ and σ , are referred

to as expected return and volatility of an asset, defined as

as well as lower volatility Lower volatility means lower investment risk

The excess expected return of an asset is the difference between expected return of the asset and the return of a risk-free asset, rf, e.g., a treasury bill(Section 2.2.6) Ratio of the excess expected return to the volatility of an

asset is called as the Sharpe ratio, defined as

SR= μ − rf

The higher the Sharpe ratio for an asset the higher the expected return for agiven volatility

Investors may further lower their risk via diversification, i.e., investing in

a portfolio of assets This practice requires us to know not only the mean and

variance of the returns of individual assets, but also the cross-correlations ofthe asset returns in the portfolio Cross-correlation (hence the covariance)

of asset returns is an important aspect of modern portfolio theory [19](Section 3.3), capital asset pricing model (Section 3.4), and relative valuemodels (Section 3.5) Hence, it plays a central role in trading strategies such

as pairs trading (Section 4.5) and statistical arbitrage (Section 4.6) The

covariance of the returns of two assets, r1(n) and r2(n) with mean values

μ1(n) and μ2(n), respectively, is defined as

in the range −1 ≤ ρ ≤ 1 For the special cases where ρ = 1, ρ = 0,

and,ρ = −1, asset returns are identical, have no correlation at all, and are

completely opposite of each other, respectively

Now, we extend the discussion into the scenario of multiple assets Weuse matrices for the ease of notation and also to employ mathematical tools

available in linear algebra The return vector of size N× 1 is defined as

where−1 is the inverse of matrix and I is the N × N identity matrix

and ˆρ12= 0.7482 Therefore, correlation and covariance matrices, ˆP and ˆC,

respectively, are found as

3.2.2 Effect of Sampling Frequency on Volatility

It follows from (3.1.25) that we can write the log-price at discrete time n as

a sum of initial log-price and all log-returns up to n as follows

where g T1(n) and g T2(n) are the log-returns associated with s T1(n) and

s T2(n), respectively, via (3.2.13) Since the summation of Gaussian random

variables is also a Gaussian random variable, if g T1(n) ∼ N (μ, σ2) then

g T2(n) ∼ N (kμ, kσ2), and

whereσ T1 andσ T2 are the standard deviation of g T1(n) and g T2(n),

respec-tively Equality given in (3.2.15) shows that volatilities at different samplingfrequencies of the same asset are related by square root of their sub-sampling

ratio k.

3.2.3 Jumps in the Returns

Geometric Brownian motion model and its improved versions with localand stochastic volatilities discussed in Section 3.1all have the continuityproperty in price However, price of an asset is impacted by many reasonsincluding asset specific and asset related business developments and finan-cial news Although some of those news are anticipated, there are manyinstances where these higher impact events happen quite randomly Weusually observe upward and downward abrupt price changes on any asset

These abrupt changes are referred to as jumps [20] One of the simplest

discrete-time price models with jumps is given as

s (n) = s(n − 1) + j(n) + ξ(n), (3.2.16)

where j (n)∈ R is the abrupt price change, up or down, that happens at discrete-time n with certain statistical model, and ξ(n) ∼ N (μ, σ2) is a

Gaussian random noise process We note that in (3.2.16), the random

log-return g (n) of (3.1.25) is modeled as the summation of two processes

Namely, a jump process j (n), and a pure Gaussian noise process ξ(n),

InFigure 3.2.1a, realization of a Gaussian random processN (μ, σ2) with

μ = 0.01 bps and σ = 2.11 bps is shown InFigure 3.2.1b, log-return of

Apple Inc (AAPL) stock on June 17, 2010 with a sampling period of T s =

5 s is displayed For this case, estimated mean (drift) and standard deviation

(b)

Figure 3.2.1 (a) A realization of a white Gaussian random process and (b) Log-returns of Apple Inc (AAPL) stock on June 17, 2010 for sampling period T = 5 s.

(volatility) of the returns are 0.01 bps and 2.11 bps, respectively We observefromFigure 3.2.1a and b that one needs to consider the jump process in themodel in order to employ the basic price model more properly Any jump ofhigh significance is the main reason for the so-called regime change in theasset price.

In order to highlight the importance of jump processes in price modeling,

we design a simple experiment as follows From (3.2.15), we define thevolatility estimation error between the two sampling intervals as follows

ε = ˆσ (m)

k /m − ˆσ(k) , (3.2.18)where ˆσ(m) and ˆσ (k) are the volatilities estimated at the intervals T s = m and T s = k, respectively, via

ˆσ (T s ) =

1

g T s (n) is the log-return of associated log-price sampled with the period

T s, ˆμ (T s ) is the estimated mean of the log-return as given

and N is the estimation window length in samples If the return process g (n)

in (3.2.17) were pure Gaussian, i.e., g (n) = ξ(n), than the error term ε

would be zero in accordance with (3.2.15) We employ a histogram basedprice jump detector where a return is labeled as a jump if its absolute value

is larger than four times the estimated volatility, i.e., 4ˆσ Next, we define an

artificial “jump-free” return process as

ˆg(n) = ˆξ(n) = g(n) − ˆj(n). (3.2.21)Then, we calculate the volatility estimation error (3.2.18) for various

sampling intervals spanning from k = 1 s to 300 s with m = 1 for both log-return and jump-free log-return of AAPL on day June 17, 2010, i.e., g (n)

andˆg(n) defined in (3.2.17) and (3.2.21), respectively We calculate the errordefined in (3.2.18) as a function of sampling interval k, ε(k), and display it

inFigure 3.2.2 We observe from the figure that removing jumps reduces thevolatility estimation error The jump process is an important phenomenon inthe price formation, and one needs to take these abrupt changes into accountfor a better model See, e.g., [20] for further details on jumps in asset returns

0 50 100 150 200 250 300 0

2 4 6 8

k

e1

e2

Figure 3.2.2 Volatility estimation error ε versus sampling period k with m = 1 as defined in (3.2.18) for real and

artificial (jump-free) returns of (3.2.17) and (3.2.21), i.e., ε1and ε2, respectively.

3.3 MODERN PORTFOLIO THEORY

Modern portfolio theory (MPT) [19] provides a framework to createefficient portfolios with the minimized risk for a given expected return

by optimally allocating the total investment capital among assets of theportfolio Before providing the details of MPT, we first define the returnand risk of portfolios

3.3.1 Portfolio Return and Risk

Let us start with two-asset portfolio and extend the discussion for the case

where n is the discrete time variable, q i (n) is the ratio of the capital invested

in the ith asset, and r i (n) is the return of the ith asset defined in (3.2.1)

The investment amount, q i (n) in (3.3.1), can be dimensionless or its unit

may be a currency We omit the time index n in the following discussions

for simplicity, noting that each term in an equation is a function of discretetime Expected return of the two-asset portfolio is calculated as

μp = E rp = q1E {r1} + q2E {r2} (3.3.2)